Patchworking real algebraic varieties
oleg.viro@@gmail.com
1991 Mathematics Subject Classification:
14G30, 14H99; Secondary 14H20, 14N10Introduction
This paper is a translation of the first chapter of my dissertation 11This is not a Ph D., but a dissertation for the degree of Doctor of Physico-Mathematical Sciences. In Russia there are two degrees in mathematics. The lower, degree corresponding approximately to Ph D., is called Candidate of Physico-Mathematical Sciences. The high degree dissertation is supposed to be devoted to a subject distinct from the subject of the Candidate dissertation. My Candidate dissertation was on interpretation of signature invariants of knots in terms of intersection form of branched covering spaces of the 4-ball. It was defended in 1974. which was defended in 1983. I do not take here an attempt of updating.
The results of the dissertation were obtained in 1978-80, announced in [Vir79a]; [Vir79b]; [Vir80], a short fragment was published in detail in [Vir83a] and a considerable part was published in paper [Vir83b]. The later publication appeared, however, in almost inaccessible edition and has not been translated into English.
In [Vir89] I presented almost all constructions of plane curves contained in the dissertation, but in a simplified version: without description of the main underlying patchwork construction of algebraic hypersurfaces. Now I regard the latter as the most important result of the dissertation with potential range of application much wider than topology of real algebraic varieties. It was the subject of the first chapter of the dissertation, and it is this chapter that is presented in this paper.
In the dissertation the patchwork construction was applied only in the case of plane curves. It is developed in considerably higher generality. This is motivated not only by a hope on future applications, but mainly internal logic of the subject. In particular, the proof of Main Patchwork Theorem in the two-dimensional situation is based on results related to the three-dimensional situation and analogous to the two-dimensional results which are involved into formulation of the two-dimensional Patchwork Theorem. Thus, it is natural to formulate and prove these results once for all dimensions, but then it is not natural to confine Patchwork Theorem itself to the two-dimensional case. The exposition becomes heavier because of high degree of generality. Therefore the main text is prefaced with a section with visualizable presentation of results. The other sections formally are not based on the first one and contain the most general formulations and complete proofs.
In the last section another, more elementary, approach is expounded. It gives more detailed information about the constructed manifolds, having not only topological but also metric character. There, in particular, Main Patchwork Theorem is proved once again.
I am grateful to Julia Viro who translated this text.
1. Patchworking plane real algebraic curves
This Section is introductory. I explain the character of results staying in the framework of plane curves. A real exposition begins in Section 2. It does not depend on Section 1. To a reader who is motivated enough and does not like informal texts without proofs, I would recommend to skip this Section.
1.1. The case of smallest patches
We start with the special case of the patchworking. In this case the patches are so simple that they do not demand a special care. It purifies the construction and makes it a straight bridge between combinatorial geometry and real algebraic geometry.
1.1.A Initial Data.
Let
See Figure 1.
For
The following construction associates with Initial Data 1.1.A above a piecewise linear curve in the projective plane.
1.1.B Combinatorial patchworking.
Take the square
If a triangle of the triangulation
Let us introduce a supplementary assumption:
the triangulation
In fact, to stay in the frameworks of algebraic geometry we need
to accept an additional assumption: a
function
1.1.C Polynomial patchworking.
Given Initial Data
and consider it as a
one-parameter family of polynomials: set
1.1.D Patchwork Theorem.
Let
Then there exists
Example 1.1.E .
Construction of a curve of degree 2 is shown in Figure 3. The broken line corresponds to an ellipse. More complicated examples with a curves of degree 6 are shown in Figures 4, 5.
For more general version of the patchworking we have to prepare patches. Shortly speaking, the role of patches was played above by lines. The generalization below is a transition from lines to curves. Therefore we proceed with a preliminary study of curves.
1.2. Logarithmic asymptotes of a curve
As is
known since Newton’s works (see [New67]), behavior of a curve
For a set
For a set
The complement of the coordinate axes in
Denote by
A polynomial in two variables is said to be quasi-homogeneous if its Newton polygon is a segment. The
simplest real quasi-homogeneous polynomials are binomials of the form
It is clear that any real quasi-homogeneous polynomial in 2 variables
is decomposable into a product of binomials of the type described
above and trinomials without zeros in
A real polynomial
For a side
The assertion in the beginning of this Section about behavior of a curve nearby the coordinate axes and at infinity can be made now more precise in the following way.
1.2.A .
Let
Theorem generalizing this proposition is formulated in Section 6.3 and proved in Section 6.4. Here we restrict ourselves to the following elementary example illustrating 1.2.A .
Example 1.2.B .
Consider the polynomial
1.3. Charts of polynomials
The notion of a chart of a polynomial is fundamental for what follows. It is introduced naturally via the theory of toric varieties (see Section 3). Another definition, which is less natural and applicable to a narrower class of polynomials, but more elementary, can be extracted from the results generalizing Theorem 1.2.A (see Section 6). In this Section, first, I try to give a rough idea about the definition related with toric varieties, and then I give the definitions related with Theorem 1.2.A with all details.
To any convex closed polygon
Recall that for
Now define the charts for two classes of real polynomials separately.
First, consider the case of quasi-homogeneous polynomials. Let
Example 1.3.A .
In Figure 10 it is shown a curve
Now consider the case of peripherally nondegenerate polynomials
with Newton polygons having nonempty interiors. Let
A pair
-
(1)
for
i=1,…,n the pair(Γi∗, Γi∗∩υ) is a chart ofaΓi and -
(2)
for
ε ,δ=±1 there exists a homeomorphismhε,δ:D→Δ such thatυ∩Δε,δ=Sε,δ∘hε,δ∘l(Vl−1(D)∩Qε,δ(a)) andhε,δ(∂D∩Di)⊂Γi fori=1,…,n .
It follows from 1.2.A that any peripherally nondegenerate real
polynomial
Example 1.3.B .
In Figure 11 it is shown a chart of
1.3.C Generalization of Example 1.3.B .
Let
be a non-quasi-homogeneous real polynomial (i. e., a real trinomial
whose the Newton polygon has nonempty interior).
For
Then the pair consisting of
Proof.
Consider the restriction of
1.3.D .
If
1.4. Recovering the topology of a curve from a chart of the polynomial
First, I shall describe an auxiliary algorithm which is a block of two main algorithms of this Section.
1.4.A Algorithm. Adjoining a side with normal vector
(α,β) .
Initial data: a chart
If
1. Drawn the lines of support of
2. Take the point belonging to
3. Cut the polygon
4. Move the pieces obtained aside from each other by parallel
translations defined by vectors whose difference is orthogonal to
5. Fill the space obtained between the pieces with a parallelogram whose opposite sides are the edges of the cut.
6. Extend the operations applied above to
7. Connect the points of edges of the cut obtained from points of
Example 1.4.B .
In Figure 12 the steps of Algorithm 1.4.A are shown. It
is applied to
Application of Algorithm 1.4.A to a chart of a polynomial
If
1.4.C Algorithm.
Recovering the topology of an affine
curve from a chart of the polynomial.
Initial data: a chart
1. Apply Algorithm 1.4.A with
2. Apply Algorithm 1.4.A with
3. Glue by
4. Glue by
5. Contract to a point all sides obtained from the sides of
6. Remove the sides which are not touched on in blocks 3, 4 and 5.
Algorithm 1.4.C turns the polygon
Example 1.4.D .
In Figure 13 the steps of Algorithm
1.4.C applying to a chart of polynomial
1.4.E Algorithm.
Recovering the topology of
a projective curve from a chart of the polynomial.
Initial data: a chart
1. Block 1 of Algorithm 1.4.C .
2. Block 2 of Algorithm 1.4.C .
3. Apply Algorithm 1.4.A with
4. Block 3 of Algorithm 1.4.C .
5. Block 4 of Algorithm 1.4.C .
6. Glue by
7. Glue by
8. Block 5 of Algorithm 1.4.C .
9. Contract to a point all sides obtained from the sides of
10. Contract to a point all sides obtained from the sides of
Algorithm 1.4.E turns polygon
1.5. Patchworking charts
Let
Example 1.5.A .
In Figure 11 and 1.5 charts
of polynomials
1.6. Patchworking polynomials
Let
Let
-
(1)
restrictions
ν|Δ(ai) are linear; -
(2)
if the restriction of
ν to an open set is linear then the set is contained in one ofΔ(ai) ; -
(3)
ν(Δ∩ℤ2)⊂ℤ .
Then
If
and say that polynomials
Example 1.6.A .
Let
Then
1.7. The Main Patchwork Theorem
A real
polynomial
1.7.A .
If
By 1.3.C , Theorem 1.7.A generalizes Theorem 1.1.D . Theorem generalizing Theorem 1.7.A is proven in Section 4.3. Here we restrict ourselves to several examples.
Example 1.7.B .
In the next Section there are a number of considerably more complicated examples demonstrating efficiency of Theorem 1.7.A in the topology of real algebraic curves.
1.8. Construction of M-curves of degree 6
One of central points of the well known 16th Hilbert’s problem [Hil01] is the problem of isotopy classification of curves of degree 6 consisting of 11 components (by the Harnack inequality [Har76] the number of components of a curve of degree 6 is at most 11). Hilbert conjectured that there exist only two isotopy types of such curves. Namely, the types shown in Figure Figure 16 (a) and (b). His conjecture was disproved by Gudkov [GU69] in 1969. Gudkov constructed a curve of degree 6 with ovals’ disposition shown in Figure 16 (c) and completed solution of the problem of isotopy classification of nonsingular curves of degree 6. In particular, he proved, that any curve of degree 6 with 11 components is isotopic to one of the curves of Figure 16.
Gudkov proposed twice — in [Gud73] and [Gud71] — simplified proofs of realizability of the third isotopy type. His constructions, however, are essentially more complicated than the construction described below, which is based on 1.7.A and besides gives rise to realization of the other two types, and, after a small modification, realization of almost all isotopy types of nonsingular plane projective real algebraic curves of degree 6 (see [Vir89]).
Construction In Figure 17 two curves of degree
6 are shown. Each of them has one singular point at which three nonsingular
branches are second order tangent to each other (i.e. this
singularity belongs to type
Choosing in the projective plane various affine coordinate systems, one obtains various polynomials defining these curves. In Figures 1.8 and 1.8 charts of four polynomials appeared in this way are shown. In 1.8 the results of patchworking charts of Figures 1.8 and 1.8 are shown. All constructions can be done in such a way that Theorem 1.7.A (see [Vir89], Section 4.2) may be applied to the corresponding polynomials. It ensures existence of polynomials with charts shown in Figure 1.8.
1.9. Behavior of curve Vℝℝ2(bt) as
t→0
Let
How do the domains containing the pieces of
It is curious that the family
Indeed,
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Thus the curves
The whole picture of evolution of
1.10. Patchworking as smoothing of singularities
In
the projective plane the passage from curves defined by
Example 1.10.A .
Let
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(note, that
Curves of degree 6 with eleven components of all three isotopy types can be obtained from this curve by small perturbations of the type under consideration (cf. Section 1.8). Moreover, as it is proven in [Vir89], Section 5.1, nonsingular curves of degree 6 of almost all isotopy types can be obtained.
1.11. Evolvings of singularities
Let
Let
Denote by
Suppose the curves
Let
The patchworking construction for polynomials gives a wide class of evolvings whose charts can be created by the modification of Theorem 1.7.A formulated below.
Let
Theorem 1.7.A cannot be applied in this situation because the
polynomial
1.11.A Local version of Theorem 1.7.A .
Under the conditions above
perturbation
An evolving of a germ, constructed along the scheme above, is called a patchwork evolving.
If
2. Toric varieties and their hypersurfaces
2.1. Algebraic tori Kℝn
In the rest of this
chapter
For
Below this variety, side by side with the affine space
Denote by
Put
Denote by
It is clear that this is a diffeomorphism.
Being Abelian group,
For
For
i. e. the translation by
For
2.1.A .
Diffeomorphism
In particular,
A hypersurface of
If
2.1.B .
Let
Proof.
Since
and therefore
Proposition 2.1.B is equivalent, as it follows from 2.1.A , to
the assertion that under hypothesis of 2.1.B the set
The following proposition can be proven similarly to 2.1.B .
2.1.C .
Under the hypothesis of 2.1.B a
hypersurface
In other words, under the hypothesis of 2.1.B the
hypersurface
-
(1)
points
((−1)w1x1,…,(−1)wnxn) withw∈ℤn ,w⊥Γ , ifK=ℝ , -
(2)
points
(eiw1x1,…,eiwnxn) withw∈ℝn ,w⊥Γ , ifK=ℂ .
2.2. Polyhedra and cones
Below by a polyhedron we mean closed convex polyhedron lying in
The set of faces of a polyhedron
By a halfspace of vector space
The minimal face of a cone is the maximal vector subspace contained in the cone. It is called a ridge of the cone.
For
Let
For cone
These are cones, which are said to be dual to
It is clear that
2.3. Affine toric variety
Let
An affine toric variety
be a presentation of
For example, if
Projection
If
Let a cone
i.e. a regular map
In particular, for any
An action of algebraic torus
With each face
Varieties
| \begin{CD}KC_{\Gamma}(\Gamma)@>{\operatorname{in}^{*}}>{}>K\Gamma @>{% \operatorname{in}^{*}}>{}>K\Delta\end{CD} |
of embeddings form a partition of
2.4. Quasi-projective toric variety
Let
In particular,
For any polyhedron
A polyhedron
^{n}\end{CD} for all vertices
The variety
Denote by
It is permissible
polyhedron.
Evidently,
Polyhedra shown in Figure 25 define the following surfaces:
The variety
The complement
We shall say that a polyhedron
Let a polyhedron
1}}(\Gamma_{1})\hookrightarrow K\Delta_{1}\end{CD}.
Obviously, these maps commute with the embeddings, by which
One can show (see, for example, [GK73]) that for any polyhedron
2.5. Hypersurfaces of toric varieties
Let
2.5.A .
Let
Proof.
Consider
2.5.B .
Let
The proof is analogous to the proof of the previous statement.
Denote by
A Laurent polynomial
It is not difficult to prove that completely nondegenerate
L-polynomials form Zarisky open subset of the space of L-polynomials
over
2.5.C .
If a Laurent polynomial
Theorem 2.5.C allows various generalizations related with
possibilities to consider singular
2.5.D .
Let
The proof of this proposition is a fragment of the proof of Theorem
2.5.C .
3. Charts
3.1. Space ℝ+Δ
The aim of this
Subsection is to
distinguish in
If
Now let
If
If
The set
Choose a collection of points
Obviously
3.2. Charts of KΔ
The space
For a point
Define a map
Consider as an example the case of
3.3. Charts of L-polynomials
Let
For
A pair consisting of
| \begin{CD}\Delta\times U_{K}^{n}@>{h\times\operatorname{id}}>{}>{\mathbb{R}}_{% +}\Delta\times U_{K}^{n}@>{\rho}>{}>K\Delta\end{CD} |
is called a (nonreduced)
It is clear that the set
As it follows from 2.5.A , if
A nonreduced
-
(1)
it map
Γ×y withΓ∈(Δ) andy∈UnK to itself and -
(2)
its restriction to
Γ×UnK withg∈(Δ) commutes with transformationsid×S:Γ×UnK→Γ×UnK whereS∈UΓ .
In the case when
3.3.A .
Let
Proof.
Let
| \begin{CD}({\mathbb{R}}_{+}\Delta\times U_{K}^{n},\,\rho^{\prime-1}(V_{K\Delta% }(a)))@>{\rho^{\prime}}>{}>(K\Delta,V_{K\Delta}(a))\\ @V{({\mathbb{R}}_{+}s\times\operatorname{id})}V{}V@V{s}V{}V\\ ({\mathbb{R}}_{+}\Delta(a)\times U_{K}^{n},\,\rho^{-1}(V_{K\Delta(a)}(a)))@>{% \rho}>{}>(K\Delta(a),V_{K\Delta(a)}(a))\end{CD} |
appears. Here
If
4. Patchworking
4.1. Patchworking L-polynomials
Let
-
(1)
all the restrictions
ν|Δi are linear; -
(2)
if the restriction of
ν to an open set is linear then this set is contained in one ofΔi ; -
(3)
ν(Δ∩ℤn)⊂ℤ .
Remark 4.1.A .
Existence of such a function
Let
4.2. Patchworking charts
Let
4.3. The Main Patchwork Theorem
Let
4.3.A .
If L-polynomials
Proof.
Denote by
Let
| \begin{CD}\Delta(b)\times U_{K}^{n+1}@>{h\times\operatorname{id}}>{}>{\mathbb{% R}}_{+}\Delta(b)\times U_{K}^{n+1}\end{CD} |
and the natural projection
is a
The pair
which is cut out by this pair
on
Therefore the pair
is a result of patchworking
For
which maps a monomial
extending the embedding
.
The embeddings constructed in this way agree to each other and define
an embedding
The sets
By 3.3.A ,
and hence, the intersection
by some homeomorphism
Thus the pair
is a result of patchworking
and, hence, the pair
is homeomorphic to a
5. Perturbations smoothing a singularity of hypersurface
The construction of the previous Section can be interpreted as a purposeful smoothing of an algebraic hypersurface with singularities, which results in replacing of neighborhoods of singular points by new fragments of hypersurface, having a prescribed topological structure (cf. Section 1.10). According to well known theorems of theory of singularities, all theorems on singularities of algebraic hypersurfaces are extended to singularities of significantly wider class of hypersurfaces. In particular, the construction of perturbation based on patchworking is applicable in more general situation. For singularities of simplest types this construction together with some results of topology of algebraic curves allows to get a topological classification of perturbations which smooth singularities completely.
The aim if this Section is to adapt patchworking to needs of singularity theory.
5.1. Singularities of hypersurfaces
Let
By singularity of a hypersurface
The multiplicity or the Milnor number of a hypersurface
of the quotient of the formal power series ring by the ideal generated
by partial derivatives
If the singularity of
The following Theorem shows that the class of singularities of finite multiplicity of analytic hypersurfaces coincides with the class of singularities of finite multiplicity of algebraic hypersurfaces.
5.1.A Tougeron’s theorem.
(see, for example,
[AVGZ82], Section 6.3). If the singularity at
The notion of Newton polyhedron is extended over in a natural way to power
series. The Newton polyhedron
However in the singularity theory the notion
of Newton diagram occurred to be more important. The Newton
diagram
It follows from the definition of the Milnor number that, if the
singularity of
For a power series
Let the Newton diagram of the Taylor series
A power series
In the case when this takes place and the Taylor series of
Thus if the Newton diagram meets all coordinate axes and the Taylor series
of
5.2. Evolving of a singularity
Now let the function
If the family
-
(1)
for
t∈[0,t0] the sphere∂B intersectsVG(φt) only in nonsingular points of the hypersurface and only transversely, -
(2)
for
t∈(0,t0] the ballB contains no singular point of the hypersurfaceVG(φt) , -
(3)
the pair
(B,VB(φ0)) is homeomorphic to the cone over its boundary(∂B ,V∂B(φ0)) .
Then the family of pairs
Conditions 1 and 2 imply existence of a
smooth isotopy
Given germs determining the same singularity, a evolving of one of them obviously corresponds to a diffeomorphic evolving of the other germ. Thus, one may speak not only of evolvings of germs, but also of evolvings of singularities of a hypersurface.
The following three topological classification questions on evolvings arise.
5.2.A .
Up to homeomorphism, what manifolds can appear as
5.2.B .
Up to homeomorphism, what pairs can appear as
Smoothings
5.2.C .
Up to topological equivalence, what are the evolvings of a given singularity?
Obviously, 5.2.B is a refinement of 5.2.A . In turn, 5.2.C is more refined than 5.2.B , since in 5.2.C we are interested not only in the type of the pair obtained in result of the evolving, but also the manner in which the pair is attached to the link of the singularity.
In the case
In the case
By the way, if we want to get questions for
5.3. Charts of evolving
Let the Taylor series
Denote by
One can choose the isotopy
We shall call the pair
B%
\end{CD}. One can see
that the boundary
5.4. Construction of evolvings by patchworking
Let
the Taylor series
Let
Denote by
5.4.A .
Under the conditions above there exists
In the case when
We shall call the evolvings obtained by the scheme described in this Section patchwork evolvings.
6. Approximation of hypersurfaces of Kℝn
6.1. Sufficient truncations
Let
We need tubular neighborhoods mainly for formalizing a notion of
approximation of a submanifold by a submanifold. A manifold presented
as the image of a smooth section of the tubular fibration of a tubular
We shall consider the space
An
Introduce a norm in vector space of Laurent polynomials over
Let
-
(1)
U∩SVKℝn(b)=∅ , -
(2)
the set
la(U∩VKℝn(b)) lies in a tubularε -neighborhoodN (from the chosen class) ofla(VKℝn(aΓ)∖SVKℝn(aΓ)) and -
(3)
la(U∩VKℝn(b)) can be extended to the image of a smooth section of the tubular fibrationN→la(VKℝn(aΓ)∖SVKℝn(aΓ)) .
The
6.1.A .
If
Standard arguments based on Implicit Function Theorem give the following Theorem.
6.1.B .
If a set
From this the following proposition follows easily.
6.1.C .
If
In the case of
6.1.D .
If a set
The following proposition describes behavior of the
6.1.E .
Let
The proof follows from comparison of the definition of
and second, the transformation
6.2. Domains of ε -sufficiency of face-truncation
For
For
For
Let
6.2.A .
If in open set
Proof.
Let
and
6.2.B .
If the truncation
Proof.
For
for
Thus if
6.2.C .
Let
This proposition is proved similarly to 6.2.B , but with the
following difference: the reference to Theorem 6.1.D is replaced
by a reference to Theorem 6.1.C .
6.2.D .
Let
Proof.
By 6.2.C , for any face
6.3. The main Theorem on logarithmic asymptotes of hypersurface
Let
It is clear that the sets
Let
6.3.A .
For any Laurent polynomial
In particular, if
6.4. Proof of Theorem 6.3.A
Theorem 6.3.A is
proved by induction on dimension of polyhedron
If
Induction step follows obviously from the following Theorem.
6.4.A .
Let
Proof.
It is sufficient to prove that for any face
Indeed, putting
we obtain a required
extension of
First, consider the case of a face
Now consider the case of face
It is clear that there exists a ball
If
| l^{-1}(\mathfrak{N}_{\varphi(\Gamma)}(DC^{-}_{\Sigma}(\Gamma))\smallsetminus% \bigcup_{\Theta\in\mathcal{G}(\Sigma),\ \Gamma\in\mathcal{G}(\Theta)}\mathfrak% {N}_{\varphi(\Theta)}(DC^{-}_{\Sigma}(\Theta)) |
and, hence, in neighborhood of a smaller set
| l^{-1}(\mathfrak{N}_{\varphi(\Gamma)}(DC^{-}_{\Delta}(\Gamma))\smallsetminus% \bigcup_{\Theta\in\mathcal{G}(\Sigma),\ \Gamma\in\mathcal{G}(\Theta)}\mathfrak% {N}_{\varphi(\Theta)}(DC^{-}_{\Delta}(\Theta)). |
Therefore for any face
It is remained to put
6.5. Modification of Theorem 6.3.A
Below in Section 6.8 it will be more convenient to use not Theorem 6.3.A but the following its modification, whose formulation is more cumbrous, and whose proof is obtained by an obvious modification of deduction of 6.3.A from 6.4.A .
6.5.A .
For any Laurent polynomial
| l^{-1}(\mathfrak{N}_{c\varphi(\Gamma)}(DC^{-}_{\Delta}(\Gamma))\smallsetminus% \bigcup_{\Sigma\in\mathcal{G}(\Delta),\ \Gamma\in\mathcal{G}(\Sigma)}\mathfrak% {N}_{\varphi(\Sigma)}(DC^{-}_{\Delta}(\Sigma)). |
6.6. Charts of L-polynomials
Let
The pair
-
(1)
there exists a homeomorphism
h:(ClDΔ,φ(Δ)∩V)×UnK→Δ×UnK such thath((ClaDΔ,φ(Δ)∩V)×y)=Δ×y fory∈UnK ,υ=h(laVKℝn(a)∩(ClDΔ,φ(Δ)∩V)×UnK and for each face
Γ ofΔ the seth((ClDΔ,φ(Δ)∩DΔ,φ(Γ)∩V)×UnK) lies in the product of the star⋃Γ∈(Σ)Σ∈(Δ)∖{Δ}Σ ofΓ toUnK ; -
(2)
for any vector
ω∈ℝn , which is orthogonal toV and, in the case ofK=ℝ , is integer, the setυ is invariant under transformationΔ×UnK→Δ×UnK defined by formula(x, (y1, …,yn))↦(x, (eπiω1y1, …,eπiωnyn)) ; -
(3)
for each face
Γ ofΔ the pair(Γ×UnK, υ∩(Γ×UnK)) is aK -chart of Laurent polynomialaΓ .
The definition of the chart of a Laurent polynomial, which, as I
believe, is clearer than the description given here, but based on the
notion of toric completion of
The set
6.7. Structure of VKℝn(bt) with small t
Denote by
This allows to take advantage of results of the previous
Section for study of
I preface the formulation describing in detail the behavior of
Denote the Newton polyhedron
For
6.7.A .
If Laurent polynomials
Denote the gradient of restriction of
In the domain, where
6.8. Proof of Theorem 6.7.A
Put
and thus from
with respect to the chosen class of tubular neighborhoods in
6.9. An alternative proof of Theorem 4.3.A
Let
Then for
By Theorem 6.7.A there exist
Indeed, it follows from 6.7.A that there exists a homeomorphism
is
obtained in result of patchworking
| (1) |
turning
References
-
[Ati81]
M. F. Atiyah, Convexity and commuting hamiltonians, Bull. London Math. Soc. 14 (1981), 1–15.
-
[AVGZ82]
V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of differentiable maps. I, “Nauka”, Moscow, 1982 (Russian), English transl., Birkhaüser, Basel, 1985.
-
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[GK73]
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-
[GU69]
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-
[Gud71]
D. A. Gudkov, Construction of a new series of
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[Gud73]
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